Boundedness and Stability of Impulsively Perturbed Delay Differential Equations

Abstract

Suppose any solution of a linear impulsive delay differential equation $$ \dot{x} (t) + \sum_{i=1}^m A_i (t) x[h_i (t)] = 0,~t \geq 0, x(s) = 0, s < 0, $$ $$ x(τ_j +0) = B_j x(τ_j -0) + α_j, ~j=1,2, ... ,$$ is bounded for any bounded sequence $\{ α_i \}$. The conditions ensuring exponential stability of this equation are presented. The behavior of solutions of the non-homogeneous equation is analyzed.